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I'm having having understanding the difference between conditional and posterior probability.

Conditional probability:

...a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred.

Source: https://en.wikipedia.org/wiki/Conditional_probability

Posterior probability:

...the conditional probability that is assigned after the relevant evidence or background is taken into account.

Source: https://en.wikipedia.org/wiki/Posterior_probability

Are they essentially the same?

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  • $\begingroup$ Posterior probability is in a sense a Bayesian application of conditional probability disguised as likelihood, but they are not the same. Frequentist statisticians would happily use conditional probability without claiming to apply posterior probability $\endgroup$
    – Henry
    Jun 12, 2016 at 20:55
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    $\begingroup$ Looking at the descriptions of the two ideas on their wikipedia pages, it seems that Posterior probability is a special case of conditional probability, where the condition is explicitly the evidence from an experiment being used as the condition. $\endgroup$ Jun 12, 2016 at 21:04

2 Answers 2

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Posterior probability is a conditional probability, but more specifically implies the probability of a particular parameter value(s) from a given parameter space when a given set of observations (say $X_i$) have been observed. So, you could say it's the revised prior for the parameter given observed $X$.

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Conditional Probability: If E and F are 2 events associated with the same sample space of a random experiment, The conditional probability of event E given that F has occurred,i.e. P(E|F) = P(E,F)\P(F)

Similarly, The conditional probability of event F given that E has occurred,i.e. p(F|E) = P(E,F)\P(F)

Here, P(E,F) is the joint probability of E and F i.e Probability that both even E and F has occurred.

Let say we have the P(E|F), P(F) (and P(E) determined by the theorem of total probability) We can determine the probability of P(F|E) using Bayes rule and in this case, P(F|E) is called the posterior probability of event E, given conditional probability P(E|F) and prior P(F)

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